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Anomalous Behavior in a Traveler’s Dilemma?
C. Monica Capra, Jacob K. Goeree, Rosario Gomez, and Charles A. Holt*
forthcoming,American Economic Review
ABSTRACT
Kaushik Basu (1994) introduced atraveler’s dilemma in which two people mustindependently decide how much to claim for identical objects that have been lost on a flight.The airline, in an attempt to prevent fraud, agrees to pay each the minimum of the two claims,with a penalty for the high claimant and a reward for the low claimant. The Nash equilibriumpredicts the lowest possible claim no matter how small the penalty and reward. The intuition thatclaims may be high is confirmed by laboratory experiments. Average claims are inversely relatedto the size of the penalty/reward parameter, which is explained by a stochastic generalization ofthe Nash equilibrium.
JEL Classification: C72, C92.Keywords: experiments, decision error, traveler’s dilemma, logit equilibrium.
The notion of a Nash equilibrium has joined supply and demand as one of the two or
three techniques that economists instinctively try to use first in the analysis of economic
interactions. Moreover, the Nash equilibrium and closely related game-theoretic concepts are
being widely applied in other social sciences and even in biology, where evolutionary stability
often selects a subset of the Nash equilibria. Many people are uneasy about the stark predictions
of the Nash equilibrium in some contexts where the extreme rationality assumptions seem
implausible. Kaushik Basu’s (1994) "traveler’s dilemma" is a particularly convincing example
of a case where the unrelenting logic of game theory is at odds with intuitive notions about
human behavior. The story associated with the dilemma is that two travelers purchase identical
* Capra: Department of Economics, University of Arkansas, Fayetteville, AR 72701; Goeree and Holt: Departmentof Economics, University of Virginia, 114 Rouss Hall, Charlottesville, VA 22901; Gomez: Department of Economics,University of Malaga, 29013 Malaga, Spain. This project was funded in part by the National Science Foundation (SBR-9617784). We wish to thank Peter Coughlan, Rob Gilles, Susana Cabrera-Yeto, Irene Comeig, Nadège Marchand, andtwo anonymous referees for suggestions.
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antiques while on a tropical vacation. Their luggage is lost on the return trip, and the airline asks
them to make independent claims for compensation. In anticipation of excessive claims, the
airline representative announces: "We know that the bags have identical contents, and we will
entertain any claim between $2 and $100, but you will each be reimbursed at an amount that
equals theminimumof the two claims submitted. If the two claims differ, we will also pay a
reward of $2 to the person making the smaller claim and we will deduct a penalty of $2 from
the reimbursement to the person making the larger claim." Notice that, irrespective of the actual
value of the lost luggage, there is a unilateral incentive to "undercut" the other’s claim. It
follows from this logic that the only Nash equilibrium is for both to make the minimum claim
of $2. As Basu (1994) notes, this is also the unique strict equilibrium, and the only rationalizable
equilibrium when claims are discrete. When one of us recently described this dilemma to an
audience of physicists, someone asked incredulously: "Is this what economists think the
equilibrium is? If so, then we should shut down all economics departments."
The implausibility of the Nash equilibrium prediction is based on doubts that a small
penalty and/or reward can drive claims all the way to an outcome that minimizes the sum of the
players’ payoffs. Indeed, the Nash equilibrium in a traveler’s dilemma is independent of the size
of the penalty or reward. Economic intuition suggests that behavior conforms closely to the Nash
equilibrium when the penalty or reward is high, but that claims rise to the maximum level as the
penalty/reward parameter approaches $0.
This paper uses laboratory experiments to evaluate whether average claims are affected
by (theoretically irrelevant) changes in the penalty/reward parameter. The laboratory procedures
are described in section I. The second and third sections contain analyses of aggregate and
individual data. Section IV presents a learning model that is used to obtain maximum likelihood
estimates of the learning and decision error parameters. The fifth section considers behavior in
the final periods after most learning has occurred, i.e. when average claims stabilize and behavior
converges to a type of noisy equilibrium which combines a standard logit probabilistic choice
rule with a Nash-like consistency-of-actions-and-beliefs condition. The learning/adjustment and
equilibrium models are complementary, and together they are capable of explaining some key
features of the data. The final section concludes.
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I. Procedures
The data were collected from groups of 9-12 subjects, with each group participating in
a series of traveler’s dilemma games during a session that lasted about an hour and a half. This
type of experiment had not been done before, and we felt that more would be learned from
letting the penalty/reward parameter,R, vary over a wide range of values. Therefore, we used
two high values ofR ($0.50 and $0.80), two intermediate values ofR ($0.20 and $0.25), and two
low values ofR ($0.05 and $0.10). The penalty/reward parameter alternated between high and
low values in parts A and B of the experiment. For example, session 1 began withR = $0.80
in part A, which lasted for 10 periods. ThenR was lowered to $0.10 in part B.
Subjects were recruited from economics classes at the University of Virginia, with the
promise that they would be paid a $6 participation fee plus all additional money earned during
the experiment. Individual earnings ranged from about $24.00 to $44.00 for a session. We
began by reading the instructions for part A (these instructions are available from the authors on
request). Although decisions were referred to as "claims," the earnings calculations were
explained without reference to the context, i.e. without mentioning luggage, etc. In each period,
subjects would record their claim on their decision sheets, which were collected and randomly
matched (with draws of numbered ping pong balls) to determine the "other’s claim" and "your
earnings," and the sheets were then returned. Claims were required to be any number of cents
between and including 80 and 200, with decimals being used to indicate fractions of cents.
Subjects only saw the claim decision made by the person with whom they were matched in a
given period. They were told that part A would be followed by "another decision making
experiment" but were not given additional information about part B. The penalty/reward
parameter was changed in part B, and random pair-wise matchings were made as before. Part
B lasted for 10 periods, except in the first two sessions where it lasted for 5 periods.
II. Data
The part A data are summarized in Figure 1. Each line connects the period-by-period
averages of the 9-12 subjects in each group. There is a different penalty/reward parameter for
each cohort, as indicated by the labels on the right. The data plots are bounded by horizontal
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dashed lines that show the maximum and minimum claims of 200 and 80. The Nash equilibrium
prediction is 80 for all treatments.
The two highest lines in Figure 1 plot the average claims for low reward/penalty
parameters of 5 and 10 (cents). The first-period averages are close to 180, and they stay high
Figure 1. Data for Part A for Various Values of the Reward/Penalty Parameter
in all subsequent periods, well away from the Nash equilibrium. The two lowest lines represent
the average claims for the higher penalty/reward parameters of 50 and 80. Note that with these
parameters, the average claims quickly fall toward the Nash equilibrium. For intermediate
reward/penalty parameters of 20 and 25, the average claims level off at about 120 and 145
respectively. The averages in the last five periods are clearly inversely related to the magnitude
of the penalty/reward parameter, and the null hypothesis of no relation can be rejected at the one
5
percent level.1
For some sessions, the switch in treatments between parts A and B caused a dramatic
Figure 2. Average Claims for Parts A and B of Sessions 1 (dark line)and Session 2 (dashed line)
change in behavior. In the two sessions usingR = 80 andR = 10, for example, the behavior is
approximately reversed, as shown in Figure 2.2 There is some evidence of a sequence effect,
since the average claims were higher forR = 10 when this treatment came first than when it
followed theR = 80 treatment that "locked" onto a Nash equilibrium. In fact, the sequence effect
was so strong in one session, with a treatment switch fromR = 50 to R = 20, that the data did
not rise in part B after converging to the Nash outcome in part A. In all other sessions, the high
R treatment resulted in lower average claims, as shown in Table 1.
1 Of the 720 (=6!) possible ways that the 6 session averages could have been ranked, there are only 6 possibleoutcomes that are as extreme as the one observed (i.e. with zero or one reversals between adjacentR values). Under thenull hypothesis the probability of obtaining a ranking this extreme is: 6/720, so the null hypothesis can be rejected (one-tailed test) at the 1 percent level.
2 Obviously, the part B data have not settled down yet after 5 periods, and therefore we decided to extend part Bto ten periods in subsequent sessions.
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Consider again the null hypothesis of no treatment effect, under which higher average
Table 1. Average Claims in the Last Five Periods for All Sessions
Session 1 2 3 4 5 6
High R Treatment 82 99 92 82 146 170
Low R Treatment 162 186 86 116 171 196
claims are equally likely in both treatments. The alternative hypothesis is that average claims
are higher for the treatments with a low value ofR. This null hypothesis can be rejected at a 3
percent significance level using a standard Wilcoxon (signed-rank) nonparametric test. Thus the
treatment effect is significant, even though it does not affect the Nash equilibrium. Basically,
the Nash equilibrium provides good predictions for high incentives (R = 80 andR = 50), but
behavior is quite different from the Nash prediction under the treatments with low and
intermediate values ofR. In particular, as shown in Figure 1, the data for the low-R treatments
is concentrated at theoppositeend of the range of feasible decisions. The original presentation
of the traveler’s dilemma as in Basu (1994), involved two travelers with low incentives relative
to the range of choices, so in this sense, the intuition behind the dilemma is confirmed.3 To
summarize, the Nash equilibrium prediction of 80 for all treatments fails to account for the most
salient feature of the data, the intuitive inverse relationship between average claims and the
parameter that determines the relative cost of having the higher claim.
Since the Nash equilibrium works well in some contexts, what is needed is not a radically
different alternative, but rather, a generalization that conforms to Nash predictions in some
situations (e.g. with highR-values) and not in others. In addition, it would be interesting to
consider dynamic theories to explain the patterns of adjustment in initial periods when the data
have not yet stabilized. Many adjustment theories in the literature are based on the idea of
3 Basu (1994) does not claim to offer a resolution of the paradox, but he suggests several directions of attack.Loosely speaking, these approaches involve restricting individual decisions to sets,T1 and T2 for players 1 and 2respectively, where each set contains all best responses to claims in the other person’s set. Such sets may exist if opensets are allowed, or if some notion of "ill defined categories" is introduced. Without further refinement these approachesdo not predict the effects of the penalty/reward parameter on claim levels.
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movement toward a best response to previously-observed decisions. The next section evaluates
some of these adjustment theories and shows that they explain a high proportion of the directions
of changes in individual claims, but not the strong effect of the penalty/reward parameter on the
levels of average claims. Then in sections IV and V we present both dynamic and equilibrium
theories that are sensitive to the magnitude of the penalty/reward parameter.
III. Patterns of Individual Adjustment
One approach to data analysis is based on the perspective that participants react to
previous experience via what is called reinforcement learning in the psychology literature. In this
spirit, Reinhard Selten and Joachim Buchta (1994) consider a model of directional adjustment
in response to immediate past experience. The prediction of the model is that changes are more
likely to be made in the direction of what would have been a best response to others’ decisions
in the previous period. The predictions of this "learning direction theory" are, therefore,
qualitative and probabilistic. The theory is useful in that it provides the natural hypothesis that
changes in the "wrong" direction are just as likely as changes in the "right" direction. This null
hypothesis is decisively rejected for data from auctions (Selten and Buchta, 1994).
Table 2. Consistency of Claim Changes with Learning Direction Theory
R = 5 R = 10 R = 20 R = 25 R = 50 R = 80 all treatments
numbers of+, -, na
79, 21, 80 62, 15, 43 50, 7, 123 94, 23, 63 65, 12, 103 50, 3, 87 400, 81, 499
percentageof + a
79 81 88 80 84 94 83
p-valueb <.00003 <.00003 <.00003 <.00003 <.00003 <.00003 <.00003
a The percentage of "+" calculations excluded the non-applicable "na" cases.b Denotes the p-value for a one-tailed test.
To evaluate learning direction theory, we categorize all individual claims after the first
period as either being consistent with the theory, "+", or inconsistent with the theory, "-".
Excluded from consideration are cases of no change, irrespective of whether or not these are
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Nash equilibrium decisions. These cases of no change are classified as "na" (for not applicable).
Table 2 shows the data classification counts by treatment. The (79, 21, 80) entry under theR = 5
column heading, for example, means that there were 79 "+" classifications, 21 "-" classifications,
and 80 "na" classifications. The percentage given just below this entry indicates that 79 percent
of the "+" and "-" changes were actually "+". The percentages exclude the "na" cases from the
denominator. The "percentage of +" row indicates that significantly more than half of the
classifications were consistent with the learning direction theory. Therefore, the null hypothesis
of no difference can be rejected in all treatments, as indicated by the "p-value" row.
Note, however, that at least part of the success of learning direction theory may be due
to a statistical artifact if subjects’ decisions are draws from a random distribution, as described
for instance by the equilibrium model in section V. With random draws, the person who has the
lower claim in a given period is more likely to be near the bottom of the distribution, and hence
to draw a higher claim in the next period. Similarly, the person with the higher claim is more
likely to draw a lower claim in the following period. In fact, it can be shown that if claims are
drawn fromany stationary distribution, the probability is 2/3 that changes are in the direction
predicted by learning direction theory.4 But even the null hypothesis that the fraction of
predicted changes is 2/3 is rejected by our data at low levels of significance.
One feature of learning direction theory in this context is that non-critical changes in the
penalty/reward parameterR do not change the directional predictions of the theory. This is
becauseR affects the magnitude of the incentive to change one’s claim, but not the direction of
the best response. This feature is shared by several other directional best-response models of
evolutionary adjustment that have been proposed recently, admittedly in different contexts. For
example, Vincent Crawford (1995) considered an evolutionary adjustment mechanism for
coordination games that was operationalized by assuming that individuals switch to a weighted
average of their previous decision and the best response to all players’ decisions in the previous
period. This adaptive learning model, which explains some key elements of adjustments in
4 Suppose that a player’s draw,x1, is less than the other player’s draw,y. Then the probability that a next draw,x2, is higher thanx1 is given by:P[x2 > x1y> x1] = P[x2 > x1, y> x1] / P[y> x1]. The numerator is equal to the probabilitythatx1 is the lowest of three draws, which is 1/3, and the denominator is equal to the probability thatx1 is the lowest of2 draws, which is 1/2. So the relevant probability is 2/3.
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coordination game experiments, is similar to directional learning with the extent of directional
movements determined by the relative weights placed on the previous decision and on the best
response in the adjustment function. Another evolutionary formulation that is independent of the
magnitude ofR is that of imitation models in which individuals are assumed to copy the decision
of the person who made the highest payoff. With two-person matchings in a traveler’s dilemma,
the high-payoff person is always the person with the lower claim, regardless of theR parameter,
so that imitation (with a little exogenous randomness) will result in decisions that are driven to
near-Nash levels.5 To conclude, individual changes tend to be in the direction of a best response
to the other’s action in the previous period, but the strong effect of the penalty/reward parameter
on the average claims cannot be explained by directional learning, adaptive learning (partial
adjustment to a best response), and imitation-based learning.
IV. A Dynamic Learning Model with Logit Decision Error
In this section we present a dynamic model in which players use a simple counting rule
to update their (initially diffuse) beliefs about others’ claims. The modeling of beliefs is
important because people will wish to make high claims if they come to expect that others will
do the same. Although the only set of internally consistent beliefs andperfectly rationalactions
is at the unique Nash equilibrium claim of 80 for all values ofR, the costs of increasing claims
above 80 depend on the size of the penalty/reward parameter. For small values ofR such
deviations are relatively costless and some noise in decision making may result in claims that are
well above 80. As subjects encounter higher claims, the (noisy) best responses to expected
claims may become even higher. In this manner, a relatively small amount of noise may move
claims well above the Nash prediction when beliefs evolve endogenously over time in a sequence
of random matchings.
This section begins with a relatively standard experience-based model of learning that
builds on the work of Jordi Brandts and Charles A. Holt (1994), David J. Cooper, Susan Garvin,
5 Fernando Vega-Redondo (1997) and Paul Rhode and Mark Stegeman (1997) have shown that this type ofimitation dynamic will drive outputs in a Cournot model up to the Walrasian levels. The intuition behind this surprising(non-Nash) outcome is that price is the same for all firms in a Cournot model, and the firm with the highest output is theone with the greatest profit. Thus imitation with a little noise will drive outputs up until price equals marginal cost.
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and John H. Kagel (1997), Dilip Mookherjee and Barry Sopher (1997), Yan Chen and Fang-Fang
Tang (1998), Colin Camerer and Teck-Hua Ho (1998), Drew Fudenberg and David K. Levine
(1998), and others. There is clearly noise in the data, even in the final periods for some
treatments, so for estimation purposes it is necessary to introduce a stochastic element. Noisy
behavior is modeled with a probabilistic choice rule that specifies the probabilities of various
decisions as increasing functions of the expected payoffs associated with those decisions. For
low values ofR, the "mistakes" in the direction of higher claims will be relatively less costly,
and hence, more probable. Our analysis will be based on the standard logit model for which
decision probabilities are proportional to exponential functions of expected payoffs. The logit
model is equivalent to assuming that expected payoffs are subjected to shocks that have an
extreme value distribution. These errors can be interpreted either as unobserved random changes
in preferences or as errors in responding to expected payoffs.6 The logit formulation is
convenient in that it is characterized by a single error parameter, which allows the consideration
of perfect rationality in the limit as the error parameter goes to zero. Maximum likelihood
techniques will be used to estimate the error and learning parameters for the data from the
traveler’s dilemma experiment.
To obtain a tractable econometric learning model, the feasible range of claims (between
80 and 200) is divided inton = 121 intervals or categories of a cent. A choice that falls in
categoryj corresponds to a claim of 80 +j - 1 cents. Players’ initial beliefs, prior to the first
period, are represented by a uniform distribution with weights of 1/n. Hence, all categories are
thought to be equally likely in the first period.7 Players update their beliefs using a simple
6 Duncan R. Luce (1959) provides an alternative, axiomatic derivation of this type of decision rule; he showed thatif the ratio of probabilities associated with any two decisions is independent of the payoffs for all other decisions, thenthe choice probability for decisioni can be expressed as a ratio:ui/Σjuj, whereui is a "scale value" number associated withdecisioni. When scale values are functions of expected payoffs, and one adds the assumption that choice probabilitiesare unaffected by adding a constant to all expected payoffs, then it can be shown that the scale values areexponentialfunctions of expected payoffs. Therefore, any ratio-based probabilistic choice rule that generalizes the exponential formwould allow the possibility that decision probabilities would be changed by adding a constant to all payoffs. While thereis some experimental evidence that multiplicative increases in payoffs reduce noise in behavior (Vernon L. Smith andJames M. Walker, 1993, 1997) we know of no evidence that behavior is affected by additive changes, except whensubtracting a constant converts some gains into (focal) losses.
7 Alternatively, the first-period data can be used to estimate initial beliefs. The assumption of a uniform prior isadmittedly a simplification, but allows us to explain why the penalty/reward parameter has a strong effect even ondecisions in the first-period.
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counting rule that is best explained by assigning weights to all categories. Letwi(j,t) denote the
weight that playeri assigns to categoryj in period t. If, in period t, player i observes a rival’s
price that falls in themth category, playeri’s weights are updated as follows:wi(m,t+1) = wi(m,t)
+ ρ, while all other weights remain unchanged. These weights translate into belief probabilities,
Pi(j,t), by dividing the weight of each category by the sum of all weights. The model is one of
"fictitious play" in which a new observation is weighted by a learning parameter,ρ, that
determines the importance of a new observation relative to the initial prior. A low value ofρ
indicates "conservative" behavior in that new information has little impact on a player’s beliefs,
which are mainly determined by the initial prior.
Once beliefs are formed, they can be used to determine the expected payoffs of all the
options available. Since in our model each player chooses amongn possible categories, the
expected payoffs are given by the sum
whereπi(j,k) is playeri’s payoff from choosing a claim equal toj when the other player claimsk.
(1)
In a standard model of best-reply dynamics, a player simply chooses the category that
maximizes the expected payoff in (1). However, as we discussed above, the adjustments in such
a model will be independent of the magnitude of the key incentive parameter,R. We will
therefore allow players to make non-optimal decisions, or "mistakes," with the probability of a
mistake being inversely related to its severity. The specific parameterization that we use is the
logit rule, for which playeri’s decision probabilities,Di(j,t), are proportional to an exponential
function of expected payoffs:
The denominator ensures that the choice probabilities add up to 1, and µ is an error parameter
(2)
that determines the effect of payoff differences on choice probabilities. When µ is small, the
decision with the highest payoff is very likely to be selected, whereas all decisions become
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equally likely (i.e. behavior becomes purely random) in the limit as µ tends to infinity. To
summarize the key ingredients of our dynamic model: (i) players start with a uniform prior and
use a simple counting rule to update their beliefs, (ii) these beliefs determine expected payoffs
by (1), and (iii) the expected payoffs in turn determine players’ choice probabilities by (2).8
This "logit learning model" can be used to estimate the error parameter, µ, and the
learning parameter,ρ. Recall that the probability that playeri chooses a claim in thejth category
in period t is given byDi(j,t), and the likelihood function is simply the product of the decision
probabilities of the actual decisions made for all subjects and all ten periods. The maximum
likelihood estimates of the error and learning parameters of the dynamic learning model are: µ
= 10.9(0.6) andρ = 0.75(0.12), with standard errors shown in parentheses. The error parameter
is significantly different from the value of zero implied by perfect rationality, which is not
surprising in light of the clear deviations from the Nash predictions.9 If the learning parameter
were equal to 1.0, each observation of another person’s decision would be as informative as prior
information, so a value of 0.7 means that the prior information is slightly stronger than the
information conveyed in a single observation.
Given these estimates for the learning and error parameter, the dynamic learning model
can be used to simulate behavior in each treatment. Table 3 shows the relationship between
actual average claims and the predictions of the logit learning model. The first row of the table
shows the average claims by treatment for the first period. Notice that there is some evidence
8 An alternative approach would specify that the probability of a given decision is an increasing function of payoffsthat have been earned when that decision was employed in the past. Thus high-payoff outcomes are "reinforced." Thistype of boundedly rational adjustment process would also be sensitive to changes in the penalty/reward parameter in thetraveler’s dilemma, since this parameter affects the expected payoffs for all decisions. See Alvin E. Roth and Ido Erev(1995) and Erev and Roth (1997) for a simulation-based analysis of reinforcement-learning models in other contexts.
9 We also estimated the learning model for each session separately, and in all cases the error parameter estimateswere significantly different from zero, except for theR = 20 session where the program did not converge. Recall thatthis treatment was the only one with an average claim that was out of the order that corresponds to the magnitude of theR parameter. The error parameter estimates (with standard errors) forR = 5, 10, 25, 50, and 80 were 6.3 (1.0), 4.0 (1.0),16.7 (5.1), 6.8 (0.9), and 9.5 (0.7) respectively. These estimates are of approximately the same magnitude, but some ofthe differences are statistically significant at normal levels, which indicates that the learning model does not account forall of the "cohort effects." These estimates are, however, of roughly the same magnitude as those we have obtained inother contexts. Capra et. al (1998) estimate an error parameter of 8.1 in an experimental study of imperfect pricecompetition. The estimates for the Lisa Anderson and Charles A. Holt (1997) information cascade experiments implyan error parameter of about 12.5 (when payoffs are measured in cents as in this paper). McKelvey and Palfrey (1998)use the Brandts and Holt (1996) signaling game data to estimate µ = 10(they report 1/µ = 0.1).
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for a treatment effect in the very first period, where average claims are highest forR = 5 andR
Table 3. Predicted and Actual Average Claims
R = 5 R = 10 R = 20 R = 25 R = 50 R = 80
Average Claim in Period 1 180 177 131 150 155 120
Average Simulated Claimfor Period 1
171 166 156 150 118 97
Average Claim for Periods 8-10 195 186 119 138 85 81
Average Simulated Claimsfor Periods 8-10
178 170 155 148 107 85
Logit Equilibrium Prediction 183 174 149 133 95 88
Nash Equilibrium Prediction 80 80 80 80 80 80
= 10, and lowest forR = 80. These averages also pick up the trends that are apparent in Figure
1; claims rise slightly for the two low-R treatments, and claims fall for the high-R treatments.
The first period predictions of the dynamic model are based on the estimated error rate and the
assumption of uniform initial priors. These predictions are shown in the second row of Table
3. The average claims in the final three periods of the experiment are shown in the third row,
and these averages can be compared with the predicted averages that result from the dynamic
model (row 4). To summarize, the parameter estimates for the logit learning model can be used
in simulations to reproduce the inverse relationship between the penalty/reward parameter and
average claims, both in the first round and in the final rounds.
As players gain experience during the experiment, the prior information becomes
considerably less important. With more experience, there are fewer surprises on average, and
this raises the issue of what happens if decisions stabilize, as indicated by the relatively flat
trends in the final periods of part A for each treatment in Figure 1. An equilibrium is a state in
which the beliefs reach a point where the decision distributions match the belief distributions,
which is the topic of the next section.
V. A Logit Equilibrium Analysis
Recall that in the previous section’s logit learning model, playeri’s belief probabilities,
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Pi(j,t) for the jth category in periodt, are used in the probabilistic choice function (2) to calculate
the corresponding choice probabilities,Di(j,t). A symmetric logit equilibrium is a situation where
all players’ beliefs have stabilized at the same distributions, so that we can drop thei and t
arguments and simply equate the corresponding decision and belief probabilities:Di(j,t) = Pi(j,t)
= P(j) for decision categoryj. In such an equilibrium, the equations in (2) determine the
equilibrium probabilities (Richard D. McKelvey and Thomas R. Palfrey, 1995). The probabilities
that solve these equations will, of course, depend on the penalty/reward parameter, which is
desirable given the fact that this parameter has such a strong effect on the levels at which claims
stabilize in the experiments. The equilibrium probabilities will also depend on the error
parameter in (2), which can be estimated as before by maximizing the likelihood function.
Instead of being determined by learning experience, beliefs are now determined by equilibrium
consistency conditions.10
It is clear from the data patterns in Figure 1 that the process isnot in equilibrium in the
early periods, as average claims fall for some treatments and rise for others. Therefore, it would
be inappropriate to estimate the logit equilibrium model with all data as was done for the logit
learning model. We used the last three periods of data to estimate µ = 8.3 with a standard error
of 0.5. This error parameter estimate for the equilibrium model is somewhat lower than the
estimate for the logit learning model (10.9). This difference may be due to the fact that the
learning model was estimated with data from all periods, including the initial periods where
decisions show greater variability. Despite the difference in the treatment of beliefs, the logit
learning and equilibrium models have similar structures, and are complementary in the sense that
the equilibrium corresponds to the case where learning would stop having much effect, i.e. where
10 Recall that the data were categorized into discrete intervals for purposes of estimation, but the actual claimdecisions in the experiment could be any real numbers. An analysis of the effect of the penalty/reward parameterR onequilibrium claim distributions can be based on a continuous formulation in which the probabilities are replaced by acontinuous density function,f(x), with corresponding distribution function,F(x). In equilibrium, these represent player’sbeliefs about others’ claims, which determine the expected payoff from choosing a claim ofx, denoted byπe(x). Theexpected payoffs determine the density of claims via a continuous logit choice function:f(x) = k exp(πe(x)/µ), wherekis a constant of integration. This is not a closed-form solution for the equilibrium distribution, since the claim distributionaffects the expected payoff function. Nevertheless, it is possible to derive a number of theoretical properties of theequilibrium claim distribution. Simon P. Anderson, Jacob K. Goeree, and Charles A. Holt (1998c) consider a class ofauction-like games that includes the traveler’s dilemma as a special case. For this class of games, the logit equilibriumexists, is unique and symmetric across players. Moreover, it is shown that, for any µ > 0, an increase in theR parameterresults in a stochastic reduction in the claim distribution, in the sense of first-degree stochastic dominance.
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decision and belief distributions are identical.
Once the error and penalty/reward parameters are specified, the logit equilibrium equations
Figure 3. Logit Equilibrium Claim Densities (with µ = 8.3)
in (2) can be solved usingMathematica. Figure 3 shows the equilibrium probability distributions
for all treatments with µ = 8.3.11 These plots reveal a clear inverse relationship between
predicted claims and the magnitude of the penalty/reward parameter, as observed in the
experiment. In particular, notice that the noise introduced in the logit equilibrium model does
more than spread the predictions away from a central tendency at the Nash equilibrium. In fact,
for low values ofR, the claim distributions are centered well away from the Nash prediction, at
11 The density forR = 80 seems to lie below that forR = 50. What the figure does not show is that the densityfor R = 80 puts most of its mass at claims that are very close to 80 and has a much higher vertical intercept.
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the oppositeend of the range of feasible choices.
We can use the logit equilibrium model (for µ = 8.3) to obtain predictions for the last
three periods. These predictions, calculated from the equilibrium probability distributions in
Figure 3, are found in the fifth row of Table 3. A comparison of predictions for the first and
final three periods shows the same trend predictions that are observed in the data; increases in
the low-R treatments and decreases in the high-R treatments. The closeness of the predicted and
actual averages for the final three periods is remarkable. In all cases, the predictions are much
better than those of the Nash equilibrium, which is 80 for all treatments (row 6 in Table 3).12
We have no formal proof that the belief distributions in the logit learning model will converge
to the equilibrium distributions, but notice that the simulated average claims (row 4 of Table 3)
are quite close to the predicted equilibrium claims (row 5 of Table 3), even after as few as 8-10
simulated matchings.To summarize, the estimated error parameter of the logit equilibrium
model can be used to derive predicted average claims that track the salient treatment effect on
claim data in the final three periods, an effect that is not explained by the Nash equilibrium.
The logit-equilibrium approach in this section does not explain all aspects of the data.
For example, the claims in part B are generally lower when preceded by very low claims in a
competitive part-A treatment, as can be seen in Figure 2. This cross-game learning, which has
been observed in other experiments, is difficult to model, and is not surprising. After all, the
optimal decision depends on beliefs about others’ behavior, and low claims in a previous
treatment can affect these beliefs. Beliefs would also be influenced by knowing the true price
of the item that was lost in the traveler’s dilemma game. This true value might be a focal point
for claims made in early periods. Another aspect of the data that is not explained by the logit
equilibrium model is the tendency for a significant fraction of the subjects to use the same
decision as in the previous period.13 Other subjects change their decisions frequently, even
when average decisions have stabilized. There seems to be some inertia in decision making that
12 The logit equilibrium has been used to explain deviations from Nash behavior in some other matrix games(Robert W. Rosenthal, 1989; McKelvey and Palfrey, 1995; and Jack Ochs, 1995), in other games with a continuum ofdecisions, e.g., the "all-pay" auction (Anderson, Goeree, and Holt, 1998a), public goods games (Anderson, Goeree, andHolt, 1998b), and price-choice games (Lopez, 1995, and C. Monica Capra et al., 1998).
13 A similar problem arises with the mixed-strategy Nash equilibrium, which sometimes explains aggregate datadespite the fact that there is "too much" autocorrelation in individuals’ decisions.
17
is not captured by the logit model. Finally, separate estimates of the logit error parameter for
each treatment reveal some differences. However, the estimates are, with one exception, of the
same order of magnigude and are similar to estimates that we have found for other games.
VI. Conclusion
Basu’s traveler’s dilemma is of interest because the stark predictions of the unique Nash
equilibrium are at odds with most economists’ intuition about how people would behave in such
a situation. This conflict between theory and intuition is especially sharp for low values of the
penalty/reward parameter, since upward deviations from the low Nash equilibrium claims are
relatively costless. The experiment reported here is designed to exploit the invariance of the
Nash prediction with respect to changes in the penalty/reward parameter. The behavior of
financially motivated subjects confirmed our expectation that the Nash prediction would fail on
two counts: claims were well above the Nash prediction for some treatments, and average claims
were inversely related to the value of the penalty/reward parameter. Moreover, these results
cannot be explained by any theory, static or dynamic, that is based on (perfectly rational) best
responses to a previously observed claim, since the best response to a given claim is independent
of the penalty/reward parameter in the traveler’s dilemma game. In particular, the strong
treatment effects are not predicted by learning direction theory, imitation theories, or evolutionary
models that specify partial adjustments to best responses to the most recent outcome.
The Nash equilibrium is the central organizing concept in game theory, and has been for
over twenty-five years. This approach should not be discarded; it has worked well in many
contexts, and here it works well for high values of the penalty/reward parameter. Rather, what
is needed is a generalization that includes the Nash equilibrium as a special case, and that can
explain why it predicts well in some contexts and not others. One alternative approach is to
model the formation of beliefs about others’ decisions, and we implement this by estimating a
dynamic learning model in which players make noisy best responses to beliefs that evolve, using
a standard logit probabilistic choice rule. In an equilibrium where beliefs stabilize, the belief and
decision distributions are identical, although the probabilistic choice function will keep injecting
some noise into the system, as is observed in experiments with relatively "flat" incentives.
The logit equilibrium model uses the logit probabilistic choice function to determine
18
decisions, while keeping a Nash-like consistency-of-actions-and-beliefs condition. This model
Figure 4. Predicted and Actual Average Claims for the Final Three PeriodsKey: Dots represent average claims for each of the treatments.
performs particularly well in the traveler’s dilemma game, where the Nash predictions are at odds
with both data and intuition about average claims and incentive effects. Consider the results for
each treatment, as shown by the dark dots in Figure 4 that represent average claims for the final
three periods plotted above the corresponding penalty/reward parameter on the horizontal axis.
If one were to draw a freehand line through these dots, it would look approximately like the dark
curved line, which is in fact graph of the logit equilibrium prediction as a function of theR
parameter (calculated on basis of the estimated value of the logit error parameter for the
equilibrium model).14 Even the treatment reversal betweenR values of 20 and 25 seems
unsurprising given the closeness of these two treatments on the horizontal axis and the flatness
of the densities for these treatments in Figure 3. Recall that the Nash prediction is 80 (the
14 Since we estimated a virtually identical error rate for a different experiment in Capra et al. (1998), the predictionsin Figure 4 could have been derived from a different data set.
19
horizontal axis) for all treatments. Thus the data are concentrated at theoppositeend of the
range of feasible claims for low values of the penalty/reward parameter, which cannot be
explained by adding errors around the Nash prediction. These data patterns are well explained
by the logit equilibrium and learning models.
20
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Basu, Kaushik. "The Traveler’s Dilemma: Paradoxes of Rationality in Game Theory." American
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Capra, C. Monica; Goeree, Jacob K.; Gomez, Rosario and Holt, Charles A."Learning and
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Fudenberg, Drew and Levine, David K. Learning in Games. Cambridge, Massachusetts: MIT
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22
Appendix A: Instructions for Part A
You are going to take part in an experimental study of decision making. The funding forthis study has been provided by several foundations. The instructions are simple, and byfollowing them carefully, you may earn a considerable amount of money. At this time, you willbe given $6 for coming on time. All the money that you earn subsequently will be yours tokeep, and your earnings will be paid to you in cash today at the end of this experiment. We willstart by reading the instructions, and then you will have the opportunity to ask questions aboutthe procedures described.
EarningsThe experiment consists of a number of periods. In each period, you will be randomly
matched with another participant in the room. The decisions that you and the other participantmake will determine the amount earned by each of you. At the beginning of each period, youwill choose a number or "claim" between 80 and 200 cents. Claims will be made by writing theclaim on a decision sheet that is attached to these instructions. Your claim amount may be anyamount between and including 80 and 200 cents. That is, we allow fractions of cents. Theperson who you are matched with will also make a claim between and including 80 and 200cents. If the claims are equal, then you and the other person each receive the amount claimed.If the claims are not equal, then each of you receives the lower of the two claims. In addition,the person who makes the lower claim earns a reward of 50 cents, and the person with the higherclaim pays a penalty of 50 cents. Thus you will earn an amount that equals the lower of the twoclaims, plus a 50 cent reward if you are the person making the lower claim, or minus a 50 centpenalty if you are the person making the higher claim. There is no penalty or reward if the twoclaims are exactly equal, in which case each person receives what they claimed.
Example: Suppose that your claim is X and the other's claim is Y.If X = Y, you get X, and theother gets Y.If X > Y, you get Y minus 50 cents, and the other gets Y plus 50 cents.If X < Y, you get X plus 50 cents, and the other gets X minus 50 cents.
Record of ResultsNow, each of you should examine the record sheet for part A. This sheet is the last one
attached to these instructions. Your identification number is written in the top-right part of thissheet. Now, please look at the columns of your record sheet for part A. Going from left toright, you will see columns for the “period,” “your claim,” “other’s claim,” “minimum claim,”penalty or reward (if any), and “your earnings.” You begin by writing down your own claim inthe appropriate column. As mentioned above, this claim must be greater than or equal to 80cents and less than or equal to 200 cents, and the claim may be any amount in this range, (i.e.fractions of cents are allowed). Use decimals to separate fractions of cents. For example, wxy.zcents indicates wxy cents, and a fraction z/10 of a cent. Similarly, xy.z cents indicates xy centsand a fraction z/10 of a cent.
After you make and record your decision for period one, we will collect all decision
23
sheets. Then we will draw numbered ping pong balls to match each of you with another person.Here we have a container with ping pong balls, each ball has one of your identification numberson it. We will draw the ping pong balls to determine who is matched with whom. After wehave matched someone with you, we will write the other’s claim, the minimum claim, the penaltyor reward, and your earnings in the relevant columns of your decision sheet and return it to you.Then, you make and record your decision for period two, we collect all decision sheets, drawping pong balls to randomly match you with another person, write the other’s claim, minimumclaim, penalty or reward, and earnings in your decision sheet and return it to you. This sameprocess is repeated a total number of ten times.
Summary
To begin, participants make and record their claims by writing the claim amount in theappropriate column of the decision sheet. Then the decision sheets are collected and participantsare randomly matched using draws of numbered ping pong balls. Once the matching is done,the other’s decision, the minimum claim, the penalty or reward, and the earnings are written oneach person’s decision sheet. Once all decision sheets are returned, participants make and recordtheir claims for the next period. The decisions determine each person’s earnings as describedabove (you will receive an amount that equals the minimum of your claim and the other person’sclaim, minus a 50 cent penalty if you have the higher claim, and plus a 50 cent reward if youhave the lower claim. There is no penalty or reward if you have the same claim as the personwith whom you are matched). Note that a new random matching is done in each period. Afterwe finish all periods, we will read to you the instructions for part B.
Final Remarks
At the end of today’s session, we will pay to you, privately in cash, the amount that youhave earned. We will add together your earnings from all parts of this exercise to determine yourtotal earnings (earnings will be rounded off to the nearest penny amount). You have alreadyreceived the $6 participation payment. Therefore, if you earn an amount X during the exercisethat follows, you will receive a total amount of $6.00 + X. Your earnings are your own business,and you do not have to discuss them with anyone.
During the experiment, you are not permitted to speak or communicate with the otherparticipants. If you have a question while the experiment is going on, please raise your hand andone of us will come to your desk to answer it. At this time, do you have any questions aboutthe instructions or procedures? If you have a question, please raise your hands and one of us willcome to your seat to answer it.
24
Appendix B: Individual Decisions, with Other’s Decision in Parentheses
Session 1: Part A withR = 80 Cents, Part B withR = 10 Cents
period S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12
1R=80
200(80)
85(140.9)
85(150)
140(120)
120(140)
100(80)
80(200)
150(85)
140.9(85)
120(140)
140(120)
80(100)
2 199.9(199)
102.3(80)
100(90)
110(85)
100(120)
80(80)
199(199.9)
90(100)
85(110)
120(100)
80(80)
80(102.3)
3 190(110)
80(80)
100(199)
80(100)
110(190)
80(90)
199(100)
85(80)
90(80)
100(80)
80(80)
80(85)
4 100(80)
80(80)
100(80)
85(80)
120(80)
80(80)
80(100)
80(100)
80(100)
100(80)
80(120)
80(85)
5 80(120)
80(80)
100(80)
80(80)
120(80)
80(80)
80(80)
80(80)
80(100)
80(80)
80(80)
80(80)
6 80(80)
80(80)
100(80)
80(80)
100(80)
80(100)
80(100)
99(80)
80(80)
80(99)
80(80)
80(80)
7 80(80)
80(80)
80(99)
80(80)
80(80)
90(80)
80(80)
99(80)
80(90)
80(80)
80(80)
80(80)
8 80(99)
80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
99(80)
80(80)
80(80)
80(80)
80(80)
9 80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
10 80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
1R=10
200(120)
95(100)
185(200)
160(80)
120(200)
100(95)
200(185)
80(160)
190(200)
200(80)
80(200)
200(190)
2 200(97)
97(200)
195(180)
170(200)
140(98)
98(140)
190(190)
100(80)
190(190)
80(100)
200(170)
180(195)
3 200(189)
105(190)
185(200)
200(125)
125(200)
200(185)
189(200)
190(105)
190(200)
80(180)
200(190)
180(80)
4 195(100)
150(189)
185(190)
160(200)
199.9(80)
190(185)
189(150)
100(195)
190(80)
80(190)
200(160)
80(199.9)
5 199(185)
175(200)
185(199)
190(180)
199(80)
200(175)
195(200)
190(190)
190(190)
80(199)
200(195)
180(190)
25
Session 2: Part A withR = 10 Cents, Part B withR = 80 Cents
period S1 S2 S3 S4 S5 S6 S7 S8 S9
1R = 10
150(199)
150(-)
199(150)
186.5(189)
190.1(190)
190(190.1)
191.3(150)
189(186.5)
150(191.3)
2 143(190)
200(179.8)
199(183.5)
154.4(-)
200(175)
190(143)
183.5(199)
179.8(200)
175(200)
3 175(141.5)
190(165)
199(175.4)
141.5(175)
190(-)
190(191.3)
191.3(190)
175.4(199)
165(190)
4 135(184.6
180(189.9)
199(175.5)
175.5(199)
190(-)
190(180)
189.9(180)
184.6(135)
180(190)
5 169(190)
180(191)
199(-)
190(180)
190(169)
190(176.7)
191(180)
176.7(190)
180(190)
6 181(189)
189(181)
199(179.4)
190.5(190)
190(190.5)
190(191)
191(190)
179.4(199)
185(-)
7 160(190)
180(200)
200(180)
165(190)
190(160)
190(165)
189(-)
189.6(185)
185(189.6)
8 190(175)
180(-)
200(188.2)
175(190)
190(190)
190(190)
189(185)
188.2(200)
185(189)
9 153(-)
180(189)
200(190)
180(180)
190(200)
190(185)
180(180)
189(180)
185(190)
10 170(179)
185(185)
199(200)
200(199)
190(190)
190(190)
179(170)
187(-)
185(185)
1R = 80
120(99.8)
89(-)
120(150)
80(109)
109(80)
170(104)
104(170)
99.8(120)
150(120)
2 80(89)
89(80)
120(-)
80(120)
80(149)
120(80)
149(80)
96.4(120)
120(96.4)
3 80.5(99)
80(101)
120(80)
99(80.5)
80(120)
90(-)
101(80)
108.9(80)
80(108.9)
4 83(118)
80(80)
120(80)
80(80)
118(83)
80(99.8)
80(120)
99.8(80)
85(-)
5 85(80)
80(99)
120(90)
80(-)
80(80)
80(80)
80(85)
99(80)
90(120)
* A dash (-) indicates a subject who remained unmatched after all others had been randomly paired.
26
Session 3: Part A withR = 50 Cents, Part B withR = 20 Cents
period S1 S2 S3 S4 S5 S6 S7 S8 S9 S10
1R=.50
199.8(99)
125(150)
197(150)
150(197)
199.9(99.9)
150(188)
188(150)
89.9(199.9)
99(199.8)
150(125)
2 199.8(100)
125(110)
174(87)
162(80)
80(162)
120(95.9)
87(174)
95.9(120)
100(199.8)
110(125)
3 99.8(150)
80(143)
148(140)
140(148)
180(99.9)
100(117)
143(80)
99.9(180)
150(99.8)
117(100)
4 99.8(115)
80(139)
139(80)
130(80)
80(130)
125(99)
99(125)
115(99.8)
99.5(124)
124(99.5)
5 99.7(80)
82(80)
90(96)
80(82)
80(99.7)
90(99)
89(89)
96(90)
99(90)
89(89)
6 99.7(80)
80(85)
80(199)
80(99.7)
80(89)
140(89.9)
199(80)
89.9(140)
89(80)
85(80)
7 80(130)
80(88.9)
198(80)
80(80)
80(80)
110(89.9)
130(80)
89.9(110)
80(198)
88.9(80)
8 80(95.9)
80(99)
80(80)
80(88.9)
80(80)
80(80)
80(80)
95.9(80)
99(80)
88.9(80)
9 80(85)
80(80)
80(90)
80(80)
80(80)
85(80)
85(80)
90(80)
80(80)
80(85)
10 82.7(80)
80(160)
80(80)
80(80)
80(80)
80(82.7)
160(80)
80(80)
80(80)
80(80)
1R=.20
99.7(124.8)
80(80)
80(80)
100(140)
80(80)
140(100)
100(198)
80(80)
198(100)
124.8(99.7)
2 99.7(80)
80(150)
80(99.7)
100(80)
80(80)
110(100)
100(110)
80(80)
150(80)
80(100)
3 99.5(110)
80(80)
80(100)
80(84.5)
80(80)
100(80)
110(99.5)
80(80)
80(80)
84.5(80)
4 99.5(80)
80(80)
80(115)
80(80)
80(80)
88(80)
115(80)
80(80)
80(88)
80(99.5)
5 99(80)
80(80)
94(80)
80(94)
80(80)
80(80)
170(80)
80(99)
80(170)
80(80)
6 80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
7 80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
8 80(80)
80(80)
80(80)
80(80)
80(80)
80(120)
80(80)
80(80)
120(80)
80(80)
9 200(80)
80(80)
80(80)
80(80)
80(200)
80(80)
80(80)
80(80)
80(80)
80(80)
10 200(80)
80(200)
80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
27
Session 4: Part A withR = 20 Cents, Part B withR = 50 Cents
period S1 S2 S3 S4 S5 S6 S7 S8 S9 S10
1R=.20
190(100)
123(200)
200(80)
80(120)
99(120)
80(200)
100(190)
200(123)
120(99)
120(80)
2 140(81)
123(150)
200(100)
100(200)
99(199)
199(99)
100(80)
150(123)
81(140)
80(100)
3 80(199)
123(100)
200(90)
119.5(150)
99(123)
199(80)
150(119.5)
123(99)
90(200)
100(123)
4 80(80)
123(200)
200(123)
125(80)
99(90)
80(125)
90(99)
80(80)
100(110)
110(100)
5 80(100)
123(80)
200(105)
80(80)
80(80)
100(80)
85(100)
80(123)
100(85)
105(200)
6 80(80)
123(180)
190(80)
80(80)
80(190)
80(180)
100(80)
80(100)
180(80)
180(123)
7 80(80)
123(80)
190(80)
80(123)
80(80)
80(190)
80(80)
80(80)
85(180)
180(85)
8 80(80)
123(190)
190(123)
80(80)
80(80)
180(180)
80(80)
80(100)
100(80)
180(180)
9 80(180)
123(100)
190(180)
80(80)
80(99)
180(80)
80(80)
99(80)
100(123)
180(190)
10 80(80)
123(119)
190(80)
80(180)
80(80)
180(80)
80(180)
80(190)
119(123)
180(80)
1R=.50
80(90)
110(80)
80(199.5)
80(80)
80(110)
199(150)
80(80)
199.5(80)
90(80)
150(199)
2 80(80)
110(175)
80(80)
80(80)
80(200)
175(110)
80(80)
80(80)
80(80)
200(80)
3 80(80)
110(80)
80(80)
80(110)
80(80)
150(80)
80(150)
80(80)
80(150)
150(80)
4 80(80)
110(150)
80(80)
80(80)
80(100)
150(110)
80(80)
80(80)
80(80)
100(80)
5 80(80)
110(80)
80(80)
80(80)
80(80)
100(80)
80(80)
80(80)
80(100)
80(110)
6 80(100)
110(80)
80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
80(110)
100(80)
7 80(80)
100(80)
80(100)
80(90)
80(80)
80(80)
80(80)
80(80)
80(80)
90(80)
8 80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
9 80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
10 80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
80(80)
28
Session 5: Part A withR = 25 Cents, Part B withR = 5 Cents
period S1 S2 S3 S4 S5 S6 S7 S8 S9 S10
1R=.25
85(95.5)
200(150)
150(200)
200(100)
190(130.2)
198.9(150)
100(200)
95.5(85)
150(98.9)
130.2(190)
2 88.5(150)
200(81.8)
165(149.8)
150(88.5)
190(125)
149.8(165)
90(135.8)
81.8(200)
125(190)
135.8(90)
3 130.5(175)
199.9(155)
155(199.9)
200(149.8)
190(98)
149.8(200)
175(125.8)
98(190)
175(130.5)
125.8(175)
4 160(150)
199.9(150)
150(100)
148(145)
190(149.8)
149.8(190)
150(199.9)
100(150)
150(160)
145(148)
5 140(140)
199.9(144.7)
145(150)
144(199.9)
190(105)
164.8(85.5)
85.5(164.8)
105(190)
150(145)
140(140)
6 145.5(125)
199.9(140)
140(199.9)
144.7(200)
200(144.7)
174.8(195)
195(174.8)
125(145.5)
150(139.5)
139.5(150)
7 140.3(194.7)
199.9(144)
145(174.8)
194.7(140.3)
144(199.9)
174.8(145)
105(125)
150.2(139.9)
125(105)
139.9(150.2)
8 148.2(140)
140(148.2)
145(194.7)
194.7(145)
144(100)
149.9(145)
89.1(135.3)
135.3(89.1)
100(144)
145(149.9)
9 139.9(95.2)
140(144.7)
145(144)
144.7(140)
144(145)
144.9(147.2)
95.2(139.9)
110(110)
110(110)
147.2(144.9)
10 139.9(112)
140(144.7)
170(140)
144.7(140)
144(175)
144.8(120)
112(139.9)
175(144)
120(144.8)
140(170)
1R=.05
140.6(140)
140(140.6)
175(199.9)
200(100)
199.9(175)
199.8(150)
150(199.8)
100(200)
180(175)
175(180)
2 139.5(92)
140(150)
175(199.9)
200(175)
199.9(175)
199.8(174.9)
92(139.5)
150(140)
175(200)
174.9(199.8)
3 139.5(199.8)
140(183)
180(180)
200(80)
199.9(153)
199.8(139.5)
80(200)
153(199.9)
180(180)
183(140)
4 140.8(180)
140(200)
180(160.2)
200(140)
199.9(199.8)
199.8(199.9)
106.2(180)
179(174.9)
180(140.8)
174.9(170)
5 160.2(200)
140(199.7)
165(97)
200(160.2)
199.7(140)
199.8(175)
97(165)
175(199.8)
175(174.9)
174.9(175)
6 170.5(174.5)
140(199.7)
170(200)
200(170)
199.7(140)
199.8(190)
103.7(172)
190(199.8)
172(103.7)
174.5(170.5)
7 170.9(173)
145(200)
175(120)
200(175)
199.7(195)
199.8(200)
200(199.8)
195(199.7)
120(175)
173(107.9)
8 172.1(194.9)
145(150)
160(200)
200(160)
194.9(172.1)
199.8(170)
85(199.2)
199.2(85)
150(145)
170(199.8)
9 173.6(194.9)
145(180)
170(195)
200(199.8)
194.9(173.6)
199.8(200)
99(140)
195(170)
140(99)
180(145)
10 186.4(200)
145(199.8)
180(194.9)
200(186.9)
194.9(180)
199.8(145)
89(120)
195(160)
120(89)
160(195)
29
Session 6: Part A withR = 5 Cents, Part B withR = 25 Cents
period S1 S2 S3 S4 S5 S6 S7 S8 S9 S10
1R=.05
150.5(200)
150(200)
200(199.9)
200(180.5)
120(199.9)
200(150)
200(150.5)
180.5(200)
199.9(120)
199.9(200)
2 195(190.1)
200(130)
185(100)
190.1(195)
130(200)
200(200)
200(199.9)
200(200)
100(185)
199.9(200)
3 189.9(199)
200(189.1)
199(189.9)
189.1(200)
135(200)
200(200)
200(135)
200(200)
140(200)
199.8(140)
4 194.5(193.1)
200(140)
186(200)
193.1(194.5)
140(200)
200(199.8)
200(186)
200(200)
200(200)
199.8(200)
5 198.9(200)
200(160)
197(200)
194.1(199.8)
160(200)
200(200)
200(200)
200(197)
200(198.9)
199.8(194.1)
6 198.9(196.2)
199(200)
188(195.1)
195.1(188)
190(200)
200(199)
200(199)
200(190)
199(200)
196.2(198.9)
7 198.9(199.9)
199.9(199.9)
195(198.8)
192(200)
190(198)
200(195)
200(192)
195(200)
198(190)
198.8(195)
8 198.9(194.9)
200(200)
190(197)
193(198)
197(190)
200(195)
200(200)
195(200)
198(193)
194.9(198.9)
9 198.9(200)
200(185)
193(200)
193(192)
185(200)
200(193)
200(198.9)
199(193.9)
192(193)
193.9(199)
10 198.9(200)
200(194)
192(196.5)
194(200)
170(192)
200(200)
200(200)
200(198.9)
192(170)
196.5(192)
1R=.25
189.9(200)
200(189.9)
200(80)
80(200)
120(185.9)
199(177)
176(199.9)
199.9(176)
177(199)
185.9(120)
2 189.9(130)
200(185)
190(80)
80(190)
130(189.9)
185(200)
176(175)
175(176)
178(185)
185(178)
3 189.9(170)
200(120)
191(175)
120(200)
150(181)
185(175)
175(185)
175(191)
181(150)
170(189.9)
4 174.9(175)
180(175)
175(174.9)
150(160)
160(150)
185(180)
175(180)
175(175)
180(185)
175(175)
5 174.9(175)
175(180)
175(174.9)
160(150)
150(160)
185(175)
175(185)
175(175)
180(175)
175(175)
6 174.9(175)
175(145)
175(175)
145(175)
150(174.9)
180(173.9)
175(174.9)
174.9(150)
175(175)
173.9(180)
7 174.9(174.9)
180(150)
175(175)
150(180)
150(175)
175(175)
175(175)
175(175)
175(150)
174.9(174.9)
8 174.8(174.8)
175(150)
175(160)
160(175)
150(175)
175(175)
174.8(174.8)
175(175)
175(174.8)
174.8(175)
9 174.5(175)
175(174.5)
173(155)
165(174.9)
155(173)
175(175)
174.5(173.9)
174.9(165)
175(175)
173.9(174.5)
10 174.4(175)
150.6(175)
175(150.6)
170(175)
155(174.8)
175(174.4)
170(172.8)
175(170)
174.8(155)
172.8(170)
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